arXiv subjects

Sofía Marlasca Aparicio

Publications and source records attributed to Sofía Marlasca Aparicio.

3 recordsLinked to original sources

arXiv2026

Nilpotent Deformation Theory via Ultrasolid Algebra

Deformation theory studies how varieties and other algebro-geometric objects vary in families. A central part of the subject is formal deformation theory, where one deforms over an Artinian base; such deformation problems are governed by Lie algebraic models. We pose the question of deforming varieties over nilpotent but not necessarily Artinian bases. These turn out to be classified by the same Lie algebraic models plus some topological structure. More precisely, we will consider partition Lie algebroids in the category of ultrasolid modules, a variation of the solid modules of Clausen and Scholze that give a well-behaved category akin to topological modules. To approach this result, we decompose deformation problems into n-nilpotent layers. Each of these layers is individually easier to understand, and is classified by simpler variants of partition Lie algebras.

math.AG↗

arXiv2026

Solid Duality for Profinite Groups

We classify profinite groups that have a Poincaré-like duality between their homology and cohomology. Our proofs work over every profinite coefficient ring, and for profinite as well as discrete coefficients. We do not only unify and generalise existing results, but also construct two novel examples of duality groups. We prove our results via the framework of condensed mathematics. This recently developed setting is a natural home for profinite objects, and our work is among the first to leverage this for the study of the (co)homology of profinite groups. Establishing our duality results involves an investigation of homological finiteness properties in condensed mathematics.

math.GR↗

arXiv2024

Ultrasolid Homotopical Algebra

Solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$, introduced by Clausen and Scholze, are a well-behaved variant of complete topological vector spaces that forms a symmetric monoidal Grothendieck abelian category. For a discrete field $k$, we construct the category of ultrasolid $k$-modules, which specialises to solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and $\mathbb{E}_\infty$ ultrasolid $k$-algebras, and their deformation theory. We focus on the subcategory of complete profinite $k$-algebras, which we prove is contravariantly equivalent to equal characteristic formal moduli problems with coconnective tangent complex.

math.AG↗